%I #5 Mar 30 2012 17:22:26
%S 2,2,4,2,2,4,2,4,2,32
%N a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists.
%C Checking k up through 1024 suggests that the sequence may continue -1, 2, 4, 2, -1, 4, 2, -1, 2, -1, 16, 2, 8, 2, 2, 4, 4, -1, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, ...
%C For any a>1 and b>1, a^k + b^k is composite for all odd k>1. Hence if n^k + (n+1)^k is prime then k must be a power of 2.
%C It is known that a(11) > 2^22. Is it possible that 11^2^m + 12^2^m is composite for all m > 0?
%H T. D. Noe, <a href="http://www.sspectra.com/math/GenFermat11.txt">Factorizations of Generalized Fermat Numbers 12^2^k + 11^2^k</a>
%e a(3)=4 because 3^2 + 4^2 = 25 is not prime, but 3^4 + 4^4 = 337 is prime.
%t lst={}; For[n=1, n<=100, n++, k=2; While[k<=2^10 && !PrimeQ[n^k+(n+1)^k], k=2*k]; If[k<=2^10, AppendTo[lst, k], AppendTo[lst, -1]]]; lst
%Y Cf. A078902.
%Y Cf. A080121.
%K hard,more,nonn
%O 1,1
%A _T. D. Noe_, Nov 14 2002